Trigonometry Vectors Calculus Derivatives Integration Limits Functions Differential Equations

# Trigonometry, Vectors, Calculus, and Rates of Change

## 1. A Connected View of Motion and Change

Trigonometry describes angles and cycles. Vectors describe magnitude and direction. Calculus describes change and accumulation. Together they form the language of motion, waves, growth, optimization, and physical trajectories.

This article assumes comfort with algebra and graphs. The central ideas are:

• A function maps inputs to outputs.
• A limit describes what a function approaches.
• A derivative measures instantaneous rate of change.
• An integral measures accumulated change.
• A differential equation states how a quantity changes.
• Vectors keep track of several directional components at once.

The best way to learn these ideas is to move among formulas, graphs, numerical tables, and real interpretations.

## 2. Angles, Degrees, and Radians

A full turn is 360 degrees or 2π radians. Radian measure is defined by arc length divided by radius:

Therefore,

Convert degrees to radians by multiplying by π/180. Convert radians to degrees by multiplying by 180/π.

Radians are natural for calculus. If a point moves around a circle of radius r with angular speed ω radians per second, its tangential speed is

This simple formula works without an extra conversion factor because the angle is measured in radians.

## 3. Right-Triangle Trigonometry

For an acute angle θ in a right triangle,

The mnemonic SOH-CAH-TOA helps recall the ratios. These ratios depend on the angle, not the triangle size, because similar triangles have proportional sides.

If a ramp rises 3 metres over a horizontal run of 4 metres, its length is 5 metres. For its angle θ,

Inverse functions recover the angle: θ = arctan(3/4).

## 4. The Unit Circle

The unit circle has radius one and centre at the origin. A point at angle θ has coordinates

This definition extends sine and cosine beyond acute angles. Signs follow the quadrant:

• Quadrant I: sine and cosine positive.
• Quadrant II: sine positive, cosine negative.
• Quadrant III: both negative.
• Quadrant IV: cosine positive, sine negative.

Because the point lies on x² + y² = 1,

Important values include:

Sine and cosine repeat every 2π. Tangent repeats every π and is undefined where cosine is zero.

## 5. Trigonometric Graphs and Models

A sinusoidal model has the form

A is amplitude, C is the midline, φ controls phase, and the period is

A daylight model, alternating voltage, pendulum approximation, or rotating wheel can all be sinusoidal. Units are essential: if x is time in seconds, ω has units radians per second.

The identity

connects angle differences to vector dot products. It also generates many other trigonometric identities.

## 6. Vectors and Components

A vector has magnitude and direction. In two dimensions,

Vector addition is componentwise:

Multiplying by scalar c changes length by |c| and reverses direction if c is negative.

The magnitude is

A unit vector has magnitude one. If u is nonzero, its direction unit vector is u/|u|.

A velocity vector separates horizontal and vertical motion. Its magnitude is speed, while its components say how position changes along each axis.

## 7. Dot Product and Angles Between Vectors

The dot product can be calculated from components:

Geometrically,

Therefore,

Perpendicular vectors have zero dot product. A positive dot product means an acute angle; a negative one means an obtuse angle.

The projection of u onto a nonzero vector v is

Work by a constant force is a dot product: W = F · displacement. Only the force component along the motion contributes.

## 8. Functions and Their Behaviour

A function f assigns an output f(x) to each input x in its domain. Important questions include:

• Where is it defined?
• Where is it zero?
• Where is it increasing or decreasing?
• Is it continuous?
• Does it repeat?
• What happens for very large inputs?

Composition feeds one function into another:

An inverse reverses a one-to-one function:

The graph of an inverse reflects across y = x. Inverse trigonometric functions use restricted domains so they are one-to-one.

## 9. Limits and Continuity

A limit describes the value f(x) approaches as x approaches a:

The function need not equal L at x = a, and it may even be undefined there. For example,

Although the original expression is undefined at 1,

A function is continuous at a if the limit exists, f(a) exists, and they are equal.

Limits also describe infinity and asymptotes. The statement f(x) → 0 as x → infinity means outputs can be made arbitrarily close to zero by taking x sufficiently large.

## 10. Derivative from First Principles

The average rate of change from x to x + h is

The derivative is the limit as the interval shrinks:

For f(x) = x²,

Taking h to zero gives

Geometrically, the derivative is tangent-line slope. Physically, position derivative is velocity and velocity derivative is acceleration. Units of a derivative are output units divided by input units.

## 11. Differentiation Rules

The power rule is

The constant multiple and sum rules let derivatives pass through linear combinations.

The product rule is

The quotient rule is

The chain rule differentiates a composition:

Example:

Common derivatives include

## 12. Tangent Lines, Motion, and Optimization

The tangent-line approximation near x = a is

Critical points occur where f′(x) = 0 or does not exist. They are candidates for maxima and minima, but must be classified by sign changes, the second derivative, or endpoint comparison.

For position s(t),

Velocity can be negative even though speed is nonnegative. A particle changes direction when velocity changes sign, not merely when acceleration is zero.

Optimization steps:

1. Define variables and the quantity to optimize.
2. Use constraints to express it with one variable.
3. Find the allowed domain.
4. Differentiate.
5. Test critical points and endpoints.
6. Interpret the answer with units.

## 13. Antiderivatives and Indefinite Integrals

An antiderivative F satisfies F′ = f. The indefinite integral represents a family:

The constant C is necessary because all vertical shifts have the same derivative.

Reverse power rule:

The special logarithm case is

Substitution reverses the chain rule. Integration by parts reverses the product rule:

## 14. Definite Integrals and Signed Area

The definite integral is a limit of sums:

It measures accumulated signed area. Regions below the x-axis contribute negatively. Total geometric area requires splitting intervals and using absolute values where needed.

If v(t) is velocity, displacement from a to b is

Integrating speed gives total distance, which may exceed the magnitude of displacement when direction changes.

## 15. Fundamental Theorem of Calculus

The Fundamental Theorem connects derivatives and integrals.

If

then, when f is continuous,

Also, if G′ = f,

The first statement says accumulation changes at the current input rate. The second turns an infinite limiting sum into evaluation of an antiderivative.

With a variable upper limit g(x), the chain rule gives

## 16. First-Order Differential Equations

A differential equation relates an unknown function to its derivatives. The exponential growth or decay model is

Separate variables:

Integrate:

Exponentiating and using y(0) = y₀ gives

Positive k gives growth; negative k gives decay. The doubling time is ln(2)/k for growth. The half-life is ln(2)/|k| for decay.

A more general first-order linear equation is

which can be solved using an integrating factor.

## 17. Projectile Trajectories

Ignoring air resistance, a projectile launched with speed v₀ at angle θ has components

Its position is

Velocity is the derivative:

Acceleration is constant and downward. The model stops being adequate when air drag, wind, Earth curvature, or changing gravity matters.

## 18. Numerical Differentiation and Integration

Computers approximate derivatives with finite differences. A centred difference is

A very large h has truncation error. A very small h suffers floating-point cancellation. Accuracy should be checked by varying h.

The trapezoid rule approximates an integral by connecting neighbouring samples with straight lines. Simpson’s rule uses quadratic pieces and can be more accurate for smooth functions.

Numerical results require convergence checks. Doubling the number of samples should change the answer less and less.

## 19. Python Calculus and Trajectory Laboratory

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad, solve_ivp

def centered_derivative(f, x, h=1e-5):
    return (f(x+h)-f(x-h))/(2*h)

f = lambda x: np.sin(x)*np.exp(-0.2*x)
x0 = 1.3
print("Numerical derivative:", centered_derivative(f, x0))
print("Analytic derivative:",
      np.exp(-0.2*x0)*(np.cos(x0)-0.2*np.sin(x0)))

integral, uncertainty = quad(f, 0.0, 4.0)
print("Integral from 0 to 4:", integral)
print("Quadrature uncertainty estimate:", uncertainty)

# Exponential ODE y' = k*y.
k = -0.7
ode = solve_ivp(lambda t, y: k*y, (0, 8), [5.0],
                dense_output=True, rtol=1e-10, atol=1e-12)
t = np.linspace(0, 8, 400)
y_numeric = ode.sol(t)[0]
y_exact = 5*np.exp(k*t)

# Projectile trajectory.
g = 9.81
speed = 24.0
angle = np.deg2rad(42.0)
vx0, vy0 = speed*np.cos(angle), speed*np.sin(angle)
flight_time = 2*vy0/g
tp = np.linspace(0, flight_time, 500)
xp = vx0*tp
yp = vy0*tp-0.5*g*tp**2
velocity = np.column_stack((np.full_like(tp, vx0), vy0-g*tp))
speed_t = np.linalg.norm(velocity, axis=1)

# Vector angle and projection example.
u = np.array([3.0, 4.0])
v = np.array([5.0, -1.0])
theta = np.arccos(np.dot(u, v)/(np.linalg.norm(u)*np.linalg.norm(v)))
projection = np.dot(u, v)/np.dot(v, v)*v
print("Angle between u and v (degrees):", np.rad2deg(theta))
print("Projection of u onto v:", projection)

fig, axes = plt.subplots(2, 2, figsize=(12, 9))
xx = np.linspace(0, 8, 500)
axes[0, 0].plot(xx, f(xx))
axes[0, 0].scatter([x0], [f(x0)])
axes[0, 0].set(xlabel="x", ylabel="f(x)", title="Function and local change")

axes[0, 1].plot(t, y_numeric, label="numerical")
axes[0, 1].plot(t, y_exact, "--", label="exact")
axes[0, 1].set(xlabel="t", ylabel="y", title="First-order ODE")
axes[0, 1].legend()

axes[1, 0].plot(xp, yp)
axes[1, 0].set(xlabel="Horizontal distance (m)", ylabel="Height (m)",
               title="Projectile trajectory")

axes[1, 1].plot(tp, speed_t)
axes[1, 1].set(xlabel="Time (s)", ylabel="Speed (m/s)",
               title="Projectile speed")
for ax in axes.flat:
    ax.grid(alpha=0.3)
plt.tight_layout()
plt.show()

## 20. Common Conceptual Errors

Degrees inside calculus formulas: Derivative identities for sine and cosine assume radians.

Dot product confused with componentwise multiplication: The dot product returns one scalar.

Derivative confused with function value: f(x) is output; f′(x) is rate.

Zero derivative assumed to mean a maximum: It may be a minimum or neither.

Integral assumed to be always positive: Definite integrals measure signed accumulation.

Forgetting +C: Indefinite integrals represent families of antiderivatives.

Treating numerical output as exact: Change step size or tolerance and test convergence.

## 21. A Study Sequence

Master the unit circle before memorizing identities. Practise vectors by drawing components. Learn derivatives from slopes and motion before using rules mechanically. Learn integrals from accumulation before building a table of antiderivatives. Connect differential equations to a story about how a quantity changes.

A strong student can move in both directions:

• formula to graph;
• graph to physical meaning;
• rate to accumulated change;
• accumulated change back to rate;
• vector components to magnitude and direction;
• numerical approximation to an error check.

That flexibility is the real foundation for later science, engineering, economics, and data analysis.

## 22. Error, Units, and Model Assumptions

Calculus answers are meaningful only when their assumptions are visible. A projectile formula assumes constant gravitational acceleration and no air resistance. Exponential population growth assumes growth rate remains proportional to population. A numerical derivative assumes samples are accurate enough and closely spaced enough.

Use dimensional analysis. If position is measured in metres and time in seconds, velocity must be metres per second and acceleration metres per second squared. An integral of velocity over time must have distance units. A derivative of energy with respect to position has force units.

Measurements also have uncertainty. Differentiation tends to amplify noise because it compares nearby values. Integration often smooths random noise but can accumulate systematic bias. Before fitting a sophisticated model, plot the raw data, examine residuals, and compare against a simple estimate.

## 23. Readiness Checklist

You are ready for multivariable calculus when you can:

• move confidently between degrees and radians;
• use the unit circle to determine signs and exact values;
• add vectors and interpret a dot product geometrically;
• explain a limit without saying that x literally reaches the limiting value;
• derive a simple derivative from the difference quotient;
• apply power, product, quotient, and chain rules;
• interpret derivatives with units;
• distinguish displacement from total distance;
• connect a definite integral to accumulation;
• use the Fundamental Theorem in both directions;
• solve dy/dt = ky and interpret the sign of k;
• compare numerical and analytic answers using an error check.

The objective is not speed alone. It is the ability to explain what a result means, what units it carries, and when the model should or should not be trusted.